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Math Help - Matrix Initial Value Problem

  1. #1
    Newbie
    Joined
    Mar 2011
    Posts
    22

    Matrix Initial Value Problem

    Hi, I'm stuck on solving this problem...
    Consider the initial value problem



    Determine the solution as a function of .

    So this is what I did so far
    Sorry I dont know how to type matrix on this....

    | 3.5 -0.75| * |x1|
    |3 -1.5| |x2|

    |3.5-
    λ -0.75|
    |3 -1.5-λ|

    λ2 - 2λ - 3 = 0
    λ = -1
    λ = 3

    λ=1
    |3.5-1 -0.75|
    |3 -1.5-1 |

    |2.5 -0.75|
    |3 0.5|

    This is where I'm stuck...I dont know what to do after

    same for
    λ=3
    |0.5 -0.75|
    |3 -4.5|

    Please help..
    Thanks!

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  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,233
    Thanks
    852

    Re: Matrix Initial Value Problem

    Quote Originally Posted by JC05 View Post
    Hi, I'm stuck on solving this problem...
    Consider the initial value problem



    Determine the solution as a function of .

    So this is what I did so far
    Sorry I dont know how to type matrix on this....

    | 3.5 -0.75| * |x1|
    |3 -1.5| |x2|

    |3.5-
    λ -0.75|
    |3 -1.5-λ|

    λ2 - 2λ - 3 = 0
    λ = -1
    λ = 3

    λ=1
    |3.5-1 -0.75|
    |3 -1.5-1 |

    |2.5 -0.75|
    |3 0.5|

    This is where I'm stuck...I dont know what to do after

    same for
    λ=3
    |0.5 -0.75|
    |3 -4.5|

    Please help..
    Thanks!

    If your eigensystem is $\large \{ (\lambda_1, \vec{v_1}), (\lambda_2, \vec{v_2}) \}$ then the solution to the system of diff eqs is given by

    $\large \begin{pmatrix}x_1(t) \\ x_2(t)\end{pmatrix} = c_1 \vec{v_1} e^{\lambda_1 t} + c_2 \vec{v_2} e^{\lambda_2 t}$

    and you use the initial conditionals to solve for $c_1$ and $c_2$.
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